3.1327 \(\int \frac{(1-2 x)^2 (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=45 \[ \frac{18 x^2}{125}-\frac{264 x}{625}-\frac{682}{3125 (5 x+3)}-\frac{121}{6250 (5 x+3)^2}+\frac{829 \log (5 x+3)}{3125} \]

[Out]

(-264*x)/625 + (18*x^2)/125 - 121/(6250*(3 + 5*x)^2) - 682/(3125*(3 + 5*x)) + (829*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0194944, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{18 x^2}{125}-\frac{264 x}{625}-\frac{682}{3125 (5 x+3)}-\frac{121}{6250 (5 x+3)^2}+\frac{829 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(2 + 3*x)^2)/(3 + 5*x)^3,x]

[Out]

(-264*x)/625 + (18*x^2)/125 - 121/(6250*(3 + 5*x)^2) - 682/(3125*(3 + 5*x)) + (829*Log[3 + 5*x])/3125

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2 (2+3 x)^2}{(3+5 x)^3} \, dx &=\int \left (-\frac{264}{625}+\frac{36 x}{125}+\frac{121}{625 (3+5 x)^3}+\frac{682}{625 (3+5 x)^2}+\frac{829}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{264 x}{625}+\frac{18 x^2}{125}-\frac{121}{6250 (3+5 x)^2}-\frac{682}{3125 (3+5 x)}+\frac{829 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0154481, size = 42, normalized size = 0.93 \[ \frac{\frac{5 \left (4500 x^4-7800 x^3-23760 x^2-17564 x-4277\right )}{(5 x+3)^2}+1658 \log (5 x+3)}{6250} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(2 + 3*x)^2)/(3 + 5*x)^3,x]

[Out]

((5*(-4277 - 17564*x - 23760*x^2 - 7800*x^3 + 4500*x^4))/(3 + 5*x)^2 + 1658*Log[3 + 5*x])/6250

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{264\,x}{625}}+{\frac{18\,{x}^{2}}{125}}-{\frac{121}{6250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{682}{9375+15625\,x}}+{\frac{829\,\ln \left ( 3+5\,x \right ) }{3125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(2+3*x)^2/(3+5*x)^3,x)

[Out]

-264/625*x+18/125*x^2-121/6250/(3+5*x)^2-682/3125/(3+5*x)+829/3125*ln(3+5*x)

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Maxima [A]  time = 1.10207, size = 49, normalized size = 1.09 \begin{align*} \frac{18}{125} \, x^{2} - \frac{264}{625} \, x - \frac{11 \,{\left (620 \, x + 383\right )}}{6250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{829}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

18/125*x^2 - 264/625*x - 11/6250*(620*x + 383)/(25*x^2 + 30*x + 9) + 829/3125*log(5*x + 3)

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Fricas [A]  time = 1.5152, size = 163, normalized size = 3.62 \begin{align*} \frac{22500 \, x^{4} - 39000 \, x^{3} - 71100 \, x^{2} + 1658 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 30580 \, x - 4213}{6250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^2/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/6250*(22500*x^4 - 39000*x^3 - 71100*x^2 + 1658*(25*x^2 + 30*x + 9)*log(5*x + 3) - 30580*x - 4213)/(25*x^2 +
30*x + 9)

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Sympy [A]  time = 0.116527, size = 36, normalized size = 0.8 \begin{align*} \frac{18 x^{2}}{125} - \frac{264 x}{625} - \frac{6820 x + 4213}{156250 x^{2} + 187500 x + 56250} + \frac{829 \log{\left (5 x + 3 \right )}}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(2+3*x)**2/(3+5*x)**3,x)

[Out]

18*x**2/125 - 264*x/625 - (6820*x + 4213)/(156250*x**2 + 187500*x + 56250) + 829*log(5*x + 3)/3125

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Giac [A]  time = 1.74329, size = 43, normalized size = 0.96 \begin{align*} \frac{18}{125} \, x^{2} - \frac{264}{625} \, x - \frac{11 \,{\left (620 \, x + 383\right )}}{6250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{829}{3125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

18/125*x^2 - 264/625*x - 11/6250*(620*x + 383)/(5*x + 3)^2 + 829/3125*log(abs(5*x + 3))